HW: Implement the following problems on WinQSB
Max 40x1+30x2
St. to: x1+2x2 <= 40
4x1+3x2 <=120
All variables are non-negative
Blue line = x1+2x2=40
Red line = 4x1+3x2=120
Corner points / X1 / X2 / 40x1+30x2 /a / 0 / 0 / 0
b / 30 / 0 / 1200
c / 24 / 8 / 1200
d / 0 / 20 / 600
Problem A: multiple solutions
When entered in QSB, it identifies the optimal solution as 30,0 with an optimal value of 1200. However, an alternate solution exists. It is 24,8 again with an optimal value of 1200. This really means there is an unlimited number of solutions. Any solution on the line between these vertices is optimal. Perhaps there was an error int eh formulation of the objective function.
Max 14x1 - 13x2
St. to: x1=4
4x1+2x2<=8
x2 >=6
All variables are non-negative
Blue line = x1=4
Red line = 4x1+2x2=8
Green line = x2=6
Problem B: infeasible solutions
When entered in QSB, the response is, that this problem is infeasible. When looking at the graph you see there are no vertices to form a feasible region. The direction of the inequality of the constraints should be checked. If OK, then the constraints should be re-evaluated. Perhaps non-negative vs. minimums of 4 and 6.
Max 4x1 + 3x2
St. to: x1=4
x2 >=2
All variables are non-negative
Blue line = x1=4
Green line = x2=2
Problem C: unbounded solutions
When entered into QSB, the response is that the problem is unbounded and it suggest to change the direction constraint C1 (X1 >= 4). When this is done (changed >= to <=) the optimal value is 20 with the solution being x1 = 4 and x2 = 2.
Another way to solve the problem is to change the objective formula from positive to negative. This is bounded and provides a optimal value of -16.
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